57
/www/nrich/html/content/98/12/bbprob2/
1
2011-02-01T00:00:01
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
<p>It's time to be giving prezzies [presents] to friends. We
usually wrap them up so that there is some fun in unwrapping them.
Little baby brothers and sisters tend to like the unwrapping more
than the present itself!</p>
<p>We should probably try to save paper, though, and use no more
than is necessary. So let's do some wrapping up and never have one
piece of paper overlapping another. Remembering this, let's have a
look at a very practical challenge.</p>
<p>It would probably be good to use some of those cubes which fit
together [multilink] that are around in most schools and put
perhaps 27 of these together to make a cube [which in this case
would be 3 by 3 by 3].</p>
<p style="text-align: center;"><mdo:image height="177" width="156" alt="Prezzy1" src="Prezzy1.jpg"></mdo:image></p>
<div style="text-align: center;"></div>
<br></br>
<p>This is just so that we can have a reasonable sized prezzie,
maybe it's a box with an interesting game in it ... I don't
know!</p>
<p>We are just concerned with its shape:-</p>
<p style="text-align: center;"><mdo:image height="178" width="159" alt="Prezzy2a" src="Prezzy2a.jpg"></mdo:image></p>
<div style="text-align: center;"></div>
<br></br>
<p>We need to wrap it up, remembering that we can have no overlaps.
One rule here is that the paper will be made up of squares the size
of the faces of the cube.</p>
<p>So you may decide on something like this:-</p>
<div style="text-align: center;"><mdo:image alt="" src="Shape1.gif"></mdo:image></div>
<br></br>
<p>You notice that I've used red paper that is red on both sides so
that nothing changes if I turn it over.</p>
<p>You could of course find other shapes to use. Here are just two
more:-</p>
<div style="text-align: center;"><mdo:image alt="" src="shape2.gif"></mdo:image> <mdo:image alt="" src="shape3.gif"></mdo:image></div>
<br></br>
<p>Now you find the rest!</p>
<p>Remember to avoid reflections or turning over!</p>
<br></br>
<p>Now for part two of this challenge. What happens if we make one
of the sides of this cube a bit different so that it becomes a
cuboid?</p>
<p style="text-align: center;"><mdo:image height="205" width="158" alt="Prezzy3a" src="Prezzy3a.jpg"></mdo:image></p>
<br></br>
<p>How many different shapes of paper can be used this time?</p>
<p>Try to find them all.</p>
<br></br>
<p>Now for the third part. What if the prezzie were different from
a cube in two directions?</p>
<p style="text-align: center;"><mdo:image height="349" width="277" src="Prezzy4.jpg" alt="Prezzy4"></mdo:image></p>
<br></br>
<p>Can you predict how many different shapes of wrapping paper
there will be this time?</p>
<br></br>
<p>If you can't, then make them all - that's one sure way of
finding out!</p>
<p>Well, good luck with this rather different challenge which only
becomes an investigation when you now ask the question "I wonder
what would happen if ...?''</p>
<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
<p class="editorial">No one has sent in a complete solution for
this problem, so please send one in if you think you have it. These
two partial solutions from Joanne and Jill might get you
thinking!!</p>
<p class="editorial">Joanne, West Flegg Middle School says:</p>
<p>I first took your plans and developed them.</p>
<p>After that I thought about what didn't work in the patterns and
I found that groups of 4 squares together in a square didn't
work:</p>
<div style="text-align: center;"><mdo:image alt="" src="t-joanne1.gif"></mdo:image> <mdo:image alt="" src="t-joanne2.gif"></mdo:image> <mdo:image alt="" src="t-joanne9.gif"></mdo:image> <mdo:image alt="" src="t-joanne10.gif"></mdo:image></div>
<p>These shapes <strong>do not</strong> work</p>
<br></br>
<p>These shapes <strong>do</strong> work. They are nets of a
cube:</p>
<div style="text-align: center;"><mdo:image alt="" src="t-joanne3.gif"></mdo:image> <mdo:image alt="" src="t-joanne4.gif"></mdo:image> <mdo:image alt="" src="t-joanne5.gif"></mdo:image> <mdo:image alt="" src="t-joanne6.gif"></mdo:image> <mdo:image alt="" src="t-joanne7.gif"></mdo:image> <mdo:image alt="" src="t-joanne8.gif"></mdo:image></div>
<br></br>
<p>I think there are 6 different ways of covering this parcel.</p>
<p><strong>Task 2:</strong> To make a net for a cuboid.</p>
<div style="text-align: center;"><mdo:image alt="" src="t-joanne12.gif"></mdo:image> <mdo:image alt="" src="t-joanne13.gif"></mdo:image></div>
<br></br>
<p>These <strong>did not</strong> work.</p>
<p>I found that you could not have the 2 end squares on the same
side.</p>
<p>These are the nets that did work:</p>
<div style="text-align: center;"><mdo:image alt="" src="t-joanne14.gif"></mdo:image> <mdo:image alt="" src="t-joanne15.gif"></mdo:image> <mdo:image alt="" src="t-joanne16.gif"></mdo:image> <mdo:image alt="" src="t-joanne17.gif"></mdo:image> <mdo:image alt="" src="t-joanne18.gif"></mdo:image> <mdo:image alt="" src="t-joanne19.gif"></mdo:image></div>
<br></br>
<p>I found 6 nets that will cover this parcel.</p>
<p>I predict that for the present that is different from a cube in
2 directions the number of nets is 3.</p>
<hr></hr>
<p class="editorial">Jill, also from West Flegg Middle School,
says:</p>
<p>First of all I found several different ways of the non net
(reflected, upside down, sideways). Here are two that I found the
most:</p>
<div style="text-align: center;"><mdo:image alt="" src="t-jill1.gif"></mdo:image> <mdo:image alt="" src="t-jill2.gif"></mdo:image></div>
<br></br>
<p>But then, when cutting out the card to test it, I cut it
wrongly.<br></br>
<mdo:image alt="" src="exclamation.gif"></mdo:image> Then a
brainwave struck me: if I move one segment over a bit it would
work!</p>
<p><mdo:image height="30" width="30" alt="" src="t-magnify.gif"></mdo:image>Here are some I
investigated:</p>
<div style="text-align: center;"><mdo:image alt="" src="t-jill3.gif"></mdo:image> <mdo:image alt="" src="t-jill4.gif"></mdo:image></div>
<br></br>
<p>But then, when I came to this one which didn't work, I realised
something: the squares that are moved have to be on the same
side.</p>
<div style="text-align: center;"><mdo:image alt="" src="t-jill5.gif"></mdo:image></div>
<br></br>
<p>This led to some interesting thoughts:</p>
<ul class="noindent">
<li>How many other nets can be made?<br></br></li>
<li>What would happen if there weren't two squares on different
sides?<br></br></li>
<li>Can we do this with other 3D shapes?</li>
</ul>
<p>Well, the truth is there must be answers out there
somewhere.</p>
<br></br>
<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><div class="embed">
<h2>Christmas Presents</h2>
<br></br>
<p>It's time to be giving prezzies [presents] to friends. We usually wrap them up so that there is some fun in unwrapping them. Little baby brothers and sisters tend to like the unwrapping more than the present itself!</p>
<p>We should probably try to save paper, though, and use no more than is necessary. So let's do some wrapping up and never have one piece of paper overlapping another. Remembering this, let's have a look at a very practical challenge.</p>
<p>It would probably be good to use some of those cubes which fit together [multilink] that are around in most schools and put perhaps 27 of these together to make a cube [which in this case would be 3 by 3 by 3].</p>
<p style="text-align: center;"><mdo:image alt="Prezzy1" height="177" src="Prezzy1.jpg" width="156"></mdo:image></p>
<div style="text-align: center;"> </div>
<br></br>
<p>This is just so that we can have a reasonable sized prezzie, maybe it's a box with an interesting game in it ... I don't know!</p>
<p>We are just concerned with its shape:-</p>
<p style="text-align: center;"><mdo:image alt="Prezzy2a" height="178" src="Prezzy2a.jpg" width="159"></mdo:image></p>
<div style="text-align: center;"> </div>
<br></br>
<p>We need to wrap it up, remembering that we can have no overlaps. One rule here is that the paper will be made up of squares the size of the faces of the cube.</p>
<p>So you may decide on something like this:-</p>
<div style="text-align: center;"><mdo:image alt="" src="Shape1.gif"></mdo:image></div>
<br></br>
<p>You notice that I've used red paper that is red on both sides so that nothing changes if I turn it over.</p>
<p>You could of course find other shapes to use. Here are just two more:-</p>
<div style="text-align: center;"><mdo:image alt="" src="shape2.gif"></mdo:image> <mdo:image alt="" src="shape3.gif"></mdo:image></div>
<br></br>
<p>Now you find the rest!</p>
<p>Remember to avoid reflections or turning over!</p>
<br></br>
<p>Now for part two of this challenge. What happens if we make one of the sides of this cube a bit different so that it becomes a cuboid?</p>
<p style="text-align: center;"><mdo:image alt="Prezzy3a" height="205" src="Prezzy3a.jpg" width="158"></mdo:image></p>
<br></br>
<p>How many different shapes of paper can be used this time?</p>
<p>Try to find them all.</p>
<br></br>
<p>Now for the third part. What if the prezzie were different from a cube in two directions?</p>
<p style="text-align: center;"><mdo:image alt="Prezzy4" height="349" src="Prezzy4.jpg" width="277"></mdo:image></p>
<br></br>
<p>Can you predict how many different shapes of wrapping paper there will be this time?</p>
<br></br>
<p>If you can't, then make them all - that's one sure way of finding out!</p>
<p>Well, good luck with this rather different challenge which only becomes an investigation when you now ask the question "I wonder what would happen if ...?''</p>
</div>
<br></br>
<h3><span style="font-weight: bold;">Why do this problem?</span></h3>
<div>This <a href="http://nrich.maths.org/public/viewer.php?obj_id=57&part=">activity</a> is very good for introducing the concept of surface area. Being very practical it helps pupils to understand the ideas before looking at any kind of formulae. It also widens the pupils' experiences of nets.</div>
<br></br>
<h3>Possible approach</h3>
<div>I would recommend that the activity is presented to the whole class as suggested with ideas explored together before the pupils go to explore on their own or in twos.</div>
<br></br>
<h3>Key questions</h3>
<div>Tell me about your shape.</div>
Have you checked it see if your paper shape works? (it's good to ask this even when you can see it's a good one!)<br></br>
<h3>Possible extension</h3>
<div>Children can alter the shape of the present, for example, a kind of cuboid shape with a hole down the middle or a cylinder.</div>
<br></br>
<h3>Possible support</h3>
<div>There may be a need for help with the pupils with fine motor skill problems.</div>
<br></br>
<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
You could always start the other way around - with the wrapped cube
- and find different ways of cutting the paper off, remembering to
keep the pieces covering each face as a square.<br></br>
Can you try to visualise wrapping the present in different ways and
then try to draw the shape of paper you needed?<br></br></mdoxml>
2
4
0
1
0
0
0
Christmas Presents
We need to wrap up this cube-shaped present, remembering that we
can have no overlaps. What shapes can you find to use?
3D Geometry, Shape and Space
Nets
3D Geometry, Shape and Space
Cubes
Mathematics Tools
Interlocking cubes
3D Geometry, Shape and Space
Cuboids
Using, Applying and Reasoning about Mathematics
Investigations